Spatio-temporal statistics is prone to the curse of dimensionality: one manifestation of this is inversion of the data–covariance matrix, which is not in general feasible for very-large-to-massive datasets, such as those observed by satellite instruments. This becomes even more of a problem in fully Bayesian statistical models, where the inversion typically has to be carried out many times in Markov chain Monte Carlo samplers. Here, we propose a Bayesian hierarchical spatio-temporal random effects (STRE) model that offers fast computation: Dimension reduction is achieved by projecting the process onto a basis-function space of low, fixed dimension, and the temporal evolution is modeled using a dynamical autoregressive model in time. We develop a multiresolutional prior for the propagator matrix that allows for unknown (random) sparsity and shrinkage, and we describe how sampling from the posterior distribution can be achieved in a feasible way, even if this matrix is very large. Finally, we compare inference based on our fully Bayesian STRE model with that based on an empirical-Bayesian STRE-model approach, where parameters are estimated via an expectation-maximization algorithm. The comparison is carried out in a simulation study and on a real-world dataset of global satellite CO2 measurements. Copyright © 2011 John Wiley & Sons, Ltd.